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So I was reading my electronic circuits textbook and am at the section of underdamped RLC unforced response, and the book mentions the natural response as

[itex]v_{n} = e^{-\alpha\cdot t}(A_{1}\cdot e^{j\omega_{d}t}+A_{2}e^{-j\omega_{d}t})[/itex]

And then after expanding the equation via Euler's formula, the text book writes

[itex]v_{n} = e^{-\alpha t}((A_{1}+A_{2})cos(\omega_{d}t)+j(A_{1}-A_{2})sin(\omega_{d}t))[/itex]

The two steps above i understand, however, the book then writes "Because the unknown constants A1 and A2 remain arbitrary, we replace (A1+A2) and j(A1-A2) with new arbitrary (yet unknown) constants B1 and B2. A1 and A2 must be complex conjugates so that B1 and B2 are real numbers, Therefore, [the above equation] becomes

[itex]v_{n} = e^{-\alpha t}(B_{1}cos(\omega_{d}t)+B_{2}sin(\omega_{d}t))[/itex]"

This is where I don't understand. Why (or how) can the imaginary portion of the equation be simply substituted for a real number?

(Book I'm using: Introduction to Electronic Circuits 8th edition (International Student Version), by Richard C. Dorf, James A. Svoboda Page 380)

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# Natural response of a RLC underdamped circuit?

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